NATTES^S , Fractical Geometry OR W TVan slated fVom the french of Le Clerc. wi th Add 1 1 1 or^s Sc Alterations i^ej /J>-^ J'uh'! as the ^/^/ nitectr thr J. C.JTeUt^.'. hy ir.MiUer.^liemartf St^j^fnji . # * J¥^TTES'S PRACTICAL GEOMETRY, OR fntrotiuction to ^crspectttje^ TRANSLATED FROM THE FRENCH OF LE CLERC; WITH ADDITIONS AND ALTERATIONS. The Explanations rendered so simple, that very young People, by Attention, may soon be enabled to go through the different Problems with perfect Ease. A WORK NOT ONLY USEFUL TO THOSE WHO CULTIVATE THE ELEGANT ART OF DRAWING, BUT ALSO RECOMMENDED TO THE STUDENT IN VARIOUS BRANCHES OF THE ARTS AND SCIENCES. TO WHICH IS ADDED, AN EASY METHOD OF MAKING AN OVAL OF ANY GIVEN PROPORTIONS: ALSO THE RULE FOR FORMING A GEOMETRICAL PLAN AND ELEVATION ; Being the last Problem previous to the Commencement of the Study of PerspectiTc, WITH FORTY VIGNETTES, BTCHID FROM DESIGNS ANALOGOUS TO THE DIFFERENT GEOMETRICAL FlCtRtS, By W. H. PYNE. jj^O\E L'e^^HE PROBLEMS ENGRAVED BY T. KING. GF THE ^> .4IVERSITY 5lU?0K^ jro^nON: PUBLISHED BY W. MILLER, ALBEMARLE STREET; AND MAY BE HAD OF MR. NATTES, NO. 5, WOODSTOCK STREET. 1805. Lf S. Go s NEIL, Printer, Little Queen Street, Holborn. INTRODUCTION. In an age like the present, when education branches out into so many studies, it is presumed he that can facilitate the acquirement of any useful science may be excused for adding another book to the general stock. The utility of Practical Geometry (at the same time that it lays a foundation for the study of Perspective, so essential to drawing local views) has rendered it a necessary part of the education of a gentleman. It forms, perhaps, the most interesting branch of mathematics, and pre- pares the mind for that species of knowledge which confers upon the pos- sessor additional consequence as a scholar. The Problems exhibited in this work are rendered so simple by the written Explanations, that the study of them cannot appear like a task ; and the youth who possesses sufficient ingenuity to invent stars for his kite, is likely to be insensibly led to draw these mathematical figures for his evening*s amusement. INDEX. Of Geometry in general Pag. X Of its Origin ib. Of its Utility ib. The Principles of Geometry 2 The Definition of a Point 4 Of a Line 4-7 Of an Angle 8 Of a Surface 11 Of rectilineal Figures ib. Of Triangles • , ib. Of quadrilateral Figures 12 Of Curves or curvilineal Figures ' ib. Of compound Figures 15 Of regular and irregular Figures ' ib. Of Axioms 18 Of Demands in order to Practice , 21 BOOK I. OF THE DESCRIPTION OF LINES. Proposition I. To erect a Perpendicular at a given Point, in the Middle of a right Line 24 IL To raise a Perpendicular at the Extremity of a given right Line ib. in. To raise a right Line upon a given Angle, so as to incline neither to the Right nor to the Left .... 2/ IV. To let fall a Perpendicular upon a given right Line, from a Point without it ...... ib. V. To draw a Line parallel to another, through a given Point , 28 VI. To bisect a given right line . . . . ib. VII. To bisect a given rectilineal Angle . . • . 31 VIII. To make an Angle equal to a given Angle at the Extremity of a Line ib. IX. To divide a given right Line into any Number of equal Parts 32 X. To draw a Tangent to a Circle from a given Point , , ib. XI. To draw a Tangent to a Circle at a given Point . . 35 XII. Given a Circle and Tangent, to find the Point of Contact , ib. A 2 VI Proposition Page Xlir, To describe a Spiral upon a given Line ... 36 XIV. Between two Points to find two others directly interposed ib. BOOK 11. OP THE CONSTRUCTION OF PLANE FIGURES. I. To make an equilateral Triangle upon a given Line . 3g II. To make a Triangle with three given right Lines . . ib. III. To make a Square upon a given right Line , . . 42 IV. To make a regular Pentagon upon a given right Line . " ib. V. To make a regular Hexagon upon a given right Line . 43 VI. To describe a Polygon of any Number of Sides, from an Hexagon to a Dodecagon, upon a given Line . . . ib. VII. To make a Polygon upon a right Line of any Number of Sides, from twelve to twenty-four . . • • . 46 VIII. To describe upon a right given Line a Segment of a Circle, capable of containing an Angle, equal to a given Angle . . ib. IX. To find the Centre of a given Circle ... 47 X. To complete a Circumference begun, whose Centre is lost . ib. XI. To describe a Circumference passing through three given Points 50 XII. To describe an Oval of a given Length . . . ib. XIII. To describe an Oval whose two Diameters are given . , 51 XIV. To find the Centre and the two Diameters of an Oval . ib. XV. To make a rectilineal Figure upon a given Line, similar to a rectili- neal Figure proposed . • • • . 54 BOOK III. OF THE INSCRIBING OF FIGURES. I. To inscribe in a given Circle an equilateral Triangle, an Hexagon, and a Dodecagon . . , , , 57 II. To inscribe a Square and an Octagon in a given Circle , 60 III. To inscribe a Pentagon or a Decagon in a given Circle . ib. IV. To inscribe an Heptagon in a given Circle . . . 61 V. To inscribe an Enneagon in a given Circle • . . ib. VI. To inscribe an Hendecagon in a given Circle . . ib. VII. To inscribe any Polygon in a given Circle ... 64 J . VIII. To cut off from a given Circle a Segment, capable of containing an ^^ Angle, equal to a rectilineal Angle proposed . . ib. Vll Proposition IX. To inscribe in a Circle, a Triangle similar to a given Triangle X. To inscribe a Circle in a given Triangle XI. To inscribe a Square in a given Triangle XII. To inscribe a regular Pentagon in an equilateral Triangle XIII. To inscribe an equilateral Triangle in a Square XIV. To inscribe an equilateral Triangle in a Pentagon XV. To inscribe a Square in a Pentagon Page 65 ib. 6s ib. 69 ib. BOOK IV. OF THE CIRCUMSCRIPTION OF FIGURES. I. To circumscribe II. To circumscribe III. To circumscribe IV. To circumscribe V. To circumscribe VI. To circumscribe VII. To circumscribe VIII. To circumscribe IX. To circumscribe X. To circumscribe a Circle about a given Triangle . • ib. a Circle about a Square ... 73 about a Circle a Triangle similar to a given Triangle ib. a Square about a Circle ... 76 a Pentagon about a given Circle , . ib. a regular Polygon about a Polygon of the same Kind 7/ a Square about an equilateral Triangle . ib. a Pentagon about an equilateral Triangle . SO a Triangle, similar to a given Triangle, about a Square ib. a Pentagon about a Square . . 81 BOOK V. OF PROPORTIONAL LINES. I. To find a mean Proportional between two others . . ib. II. Given the Sum of the Extremes and mean Proportional, to distinguish the Extremes ...... 84 III. Given the Mean of three Proportionals, and the Difference of the Extremes, to find the Extremes . . . , ib IV. To find a third Proportional to two given Lines . , 85 V. To cut off from a given right Line, a Part that shall be a mean Pro- portional between the Remainder and a third Line proposed ib. VI. To find a fourth Proportional • . . . 88 VII. To find two mean Proportionals between two given right Lines lb VIII. To cut two given Lines, each into two Parts, so that the four Seo-- ments shall be proportional ' * . . 8g IX. Given the Excess of the Diagonal of a Square above the Side, to find the Magnitude of that Side • • • . ib X. To divide a given Line in extreme and mean Proportionals . 92 VIU Proposition. Page Xr. To divide a Line according to any given Ratio • . ^2 XII. To make two Rectangle.^ upon a given Line, that shall be in a given Ratio ...... 93 XIII. To make a geometrical Plan and Elevation ... 95 XIV. Containing the Plan, Elevation, and Perspective of a double Cross 96 XV. Method of making an Oval of any given Dimension . 98 IL I S T OF THE VIGNETTES UNDER EACH PLATE, THE FRONTISPIECE^ A Mill by the Bridge at Bray, near Dublin. Plate 1 . A Horse-mill, used in a Manufactory for making Whiting. 2. Stonemasons, with the Stone-sawyer's Shed, &c. 3. A Capstan for raising Stones, Timber, and other heavy Bodies^ from Vessels on the River. 4. Wheelwrights at Work. 5. Men grinding. 6. Stonemasons laying a Pavement. 7. Copper-plate Printers at Work. 8. Figures at the Invalids Well, Paris. 9. Horses and Sheep at a Crib. 10. A Stall at a Country Fair. 1 1 . Bricklayers at Work. 12. Men weighing, the Scales suspended on a Triangle. 13. Shipwrights making a Mast. 14. A Group of Utensils for the Dairy. 15. A Windmill. 16. A Water-conduit, with a Group of Water-carts. 17. Tile-kilns. 18. Machine for removing Timber, with Woodmen at Work. 19. Gipsies cooking. 20. A Well, with a Horse raising Water. 21. Fishermen at a Capstan. 22. Gothic Conduit, with a Woman fetching Water. 23. Machine for raising Timber, &c. 24. A Pump, with a Water-cart. 25. Cooperage, with a Brewer's Vat, &c. 26. A Brickmaker's Mill for grinding Clay. 27. Sawyers at Work. 28. Market-cross, with Booths, Figures, &c. 29. An Ice-house in a Garden, with Men rolling a Gravel Walk. 30. Making a Hay- stack, with a Horse-sledge, &c. 31. Loading a Timber-tug under a Triangle. Plate 32. A Fountain in a Garden, with Steps, Vase, Figures, &c. 33. Plasterers at Work. 34. Thatching a Hay-stack. 35. A Lime-kiln. 3d. A Grecian Ruin. 37. Turners at Work. 38. Making a Rudder for a Ship of War. 39. Group of Packages, Anchor, arid Cannons. 40. A Landing-place, with Boats, &c. 41. A Steam-engine for raising Coal, &c. OF GEOMETRY IN GENERAL. The word Geometry is of Greek origin 3 in its primary sense it means the art of measuring the earth, or any distances thereon. We nevertheless understand by it the principal part of the mathematics, which is a science whose object is con- tinued quantity. Continued quantity is that, all of whose parts are conjoint, as are all sorts of extensions, sizes, and dimensions. These dimensions principally consist either in lines, angles, superficies, or bodies, considered in themselves, and without any regard to matter. Geometry is divided into speculative and practical. Speculative geometry teaches us to perceive and demonstrate the truth of geometrical propositions. Practical geometry is that which applies the theorems of speculative geometry to practice. OF ITS ORIGIN. Geometiy had its rise in Egypt, where the inundations of the Nile rendered it necessary to distinguish lands by considering their figures ; to be able to measure their respective quantities ; and to know how to lay them out in their just dimen- sions and situations. By the Egyptian studies and observations this very mechanical exercise was insensibly raised to one of the highest rank among the sciences. OF ITS UTILIIT. Geometry is not only useful, but it may be said to be absolutely necessary. It is by geometry that the astronomer makes his observations, measures the ex.- B tent of the heavens, the motions of the stars, the regularity of the seasons, and the duration of time. It is by its means that the geographer can bring into one point of view, the vv^hole earth, the vast extent of the sea, and the division of empires, kingdoms, and provinces. To it the architect is indebted for the know- ledge of the just and necessary rules for the structure of edifices both public and private. By it the engineer is enabled to construct his works : to observe the situation and take the plan of places, even when by their locality they are only accessible to the sight. To persons of the military profession it is absolutely necessaiy, not only as an introduction to fortification, by which they are taught to build ram- parts for the security of places, and to construct works for their destruction -, but it likewise gives them a considerable degree of knowledge and facility in the mili- tary art, such as drawing up an army in order of battle, marking out encamp- ments, and taking such military plans of a country as shall make them as much, esteemed for their professional knowledge as for their courage. The artist is likewise under the necessity of knowing something of geometry, as without it he cannot have such a perfect knowledge of architecture or perspec- tive as is absolutely necessary for him. Geometry is estahlished upon three Sorts of Principles, viz. Definitions, Axioms, and Petitions. Definitions are succinct explanations of names and terms. Axioms are sentences whose truth is so clear, that they are incontestable. And the petitions are such clear and intelligible demands, that the execution and practice require no demonstration. N. B. It is particularly recommended that a pupil should never go to a second proposition before he understands the first well ; and in order to practise with precision it is necessary to have good instruments : the compiler never found any maker superior to Wellington, Mathematical Instrument Maker to their Royal Highnesses the Dukes of Gloucester and Cumberland^ Crown Court, St. Ann's, Soho, London. DEFINITIONS. Ba 4 THE DEFINITION OF A POINT. A POINT is that which has no parts. By this definition you may easily perceive, that a point has neither length, nor breadth, nor depth ; that it is not any thing sensible, but only intellectual j for nothing falls under the notice of our senses that has nothing of quantity, and nothing is quantity that has not parts j so that to say a point is sensible, would be to say it has parts, which would contradict this definition. Notwithstanding, since no operation can be performed without the Intervention of something corporeal, we usually represent a mathematical point by a physical point, which is an object of sight the smallest and the least sensible that can be, and which has no geometrical magnitude divisible to our senses, and is made by the prick of a pin, point of a compass, pen, or pencil, as the point marked ......... A A central point, or centre, is a point from which a circle, or circumference, is described ; or rather it is the middle of a figure, as the point . . B A secant point, or, as some call it, a point of intersection, is a point where two or more lines cross one another, as the point , . , . C THE DEFINITION OF A LINE. A line is a length without any breadth. A line is nothing but the track made by a point passing from one place to ano- ther 5 and would not be perceived, if it were not delineated by a physical point, which by its motion represents a line to us, as . . A B, C D, E F There are as many sorts of lines as there are diflferent kinds of motions, which a point, the principle of a line, is capable of j tliough there are but two, which are simple and the principal, viz. a right and curve, and a third, which is called a mixed line, because made up of the two former, tliat are usually considered in geometry. A right line is one that lies equally between its extremities. Otherwise it is a line that goes from one point to another without any devia- tion, as . ......... A B A curve line is that which turns out of its way by one or more deviations, as C D When such a line as this is described by a pair of compasses, it is called a circu- lar line, as ....,.,,. . E A mixed line is that which is both straight and a curve, as the line V 5 FH %' K B V *- 4-' -^ 's y -^/.f D N M .r,-^' A Line is distinguished intojinite and injinite, into apparent and occult, A FINITE line is a bounded line^ containing or supposing a necessary length, as A An infinite line is an undetermined line, having no precise length, as B An apparent line is one described with ink or a pencil, as . . A B An occult, or white line, is only made with the point of a pair of compasses, or marked by points, and then it is called a pricked line, as . . Q A Line receives also several Denominations , according to its different Positions and Properties, A perpendicular is a right line that is let fall or erected upon another, making the angles on each side equal, as ...... A B A plumb line is that which hangs down without inclining to the right or left, and would pass through the centre of the earth, if it were produced infinitely, as C A horizontal line is a line in equilibrio, equally inclined on both sides D E Parallel lines are such as follow one another at an equal distance . . H An oblique line is one that is neither horizontal nor perpendicular . F G A base is a line upon which the figure rests, as . , . , I L Sides are the lines that contain a figure, as . . . I N, L M A diagonal is a right line crossing a figure, and terminated at its two opposite angles ........ ...AB A diameter is a right line passing through the centre of a circle, and termi- nated at the circumference CD A spiral line is a curve line issuing from a centre, and continually going off from it at every turn . . . . . . . . E F A chord or subtense is a right line that joins the two extremities of an arc G H An arc is any part of a circumference . . . . . G I H A tangent is a line that touches a figure without cutting it_^ nor would it cut or cross the figure, though it were produced, as , . , , L M A secant is a line that does cross or cut a figure • • . L O, M O- If two lines meet at their extremities, they meet either directly or indirectly : if directly, tliey make but one line 3 if indirectly, they form an angle. THE DEFINITION OF AN ANGLE. An angle is the indirect concourse of two lines in the same point ; or rather it is the space contained between the indirect concourse of two lines meeting in a point, as • . . . . . . . . .ABC If the concourse be formed by two right lines, the angle is called a rectilineal 5 if by two curve lines, a curvilineal j but if by one right and one curve line, a mixti- lineal angle. A denotes a rectilineal angle. B a curvilineal angle. C a mixtilineal or compound angle. A rectilineal angle receives several particular names according as it has a greater or less aperture, as right, acute, obtuse : thus the terms of rectilineal, curvilineal, and mixed, express the quality of the lines, and those of right, acute, obtuse, the quantity of the space contained between the said lines. An angle is right, when one of the lines is perpendicular to tlie other E D F An angle is acute, when its aperture is less than that of a right angle E D G An angle is obtuse, when its aperture is greater than that of a right angle F D G The middle letter D denotes the angle. ^ ^ ns M L B B "f^Sy^r-^ *^_i-/^ ^^\^^^ ^ >^K. «w^ IE r 1 . m i 'ills )5^x<<::;;r pFTHE J fo B <^1H| /B m \ lie Pl4 11 DEFINITION OF A SURFACE. A Surface is whatever has Length and Breadth without Depth or Thickness. According to the sentiments of the geometricians, a surface is a production of a hne, just as a line is the production of a point : thus we are to imagine the line EF moving towards GH to constitute the surface EF GH, which is an ex- tension bounded by lines, and has length and breadth without depth or thickness : this is commonly called a surface j but a figure, if it be considered in regard of its extremities, which are the bounding lines. If the surface be elevated or raised, it is said to be convex j but if depressed^ sunk in, or hollow, it is called a concave} and if even and flat, a plane. Thus, B is a convex surface. C a concave surface. A a plane surface. D a surface that is convex, concave, and plane. This first part relates only to plane surfaces. The terminus, term, or boundary, of any thing is its extremity : thus a point is the terminus of a line, a line is tlie term of a surface, and a surface is tlie terminus of a body. OF SURFACES OR FIGURES THAT ARE RECTILINEAL. Surfaces take their particular Names from the Numher of their Sides. Thus, A is a trigon or triangle, a figure with three sides. B a tetragon or square, a figure of four sides. C a pentagon, or a figure of five sides. D an hexagon, or figure of six sides. E an heptagon, or figure of seven sides. F an octagon, or figure of eight sides. G an enneagon, or figure of nine sides. H a decagon, or figure of ten sides. I an hendecagon, or figure of eleven sides. L a dodecagon, or figure of twelve sides. All these figures are also called by the general name of polygons. OF TRIANGLES. Triangles are distinguished by the Quality of their Angles ^ and ly the Disposition of their Sides. Thus, M is a right-angled triangle^ i.e. has one right angle. N an obtuse-angled triangle; i. e. has one obtuse angle. O an acute-angled triangle; i. e. has all three angles acute. P an equilateral triangle; i.e. has its three sides equal. Q. an isosceles triangle; i. e. has only two sides equal. R a scalene triangle; i. e. has all its three sides unequal. 12 OF QUADRILATERALS, OR FIGURES THAT HAVE FOUR SIDES. A is a square, or figure that has its four sides equal and four angles right. B, a rectangle, by some improperly called a long square, has all its angles right or equal, but its sides unequal. C, a rhombus, is a quadrilateral that has its four sides equal, but not its four angles. D, a rhomboid, has the opposite angles and sides equal, without being equian- gular or equilateral. A B C D, a parallelogram, is a quadrilateral whose opposite sides are parallel. E, a trapezium, has only two opposite sides parallel, and the two others equal. F, a trapezoid, has its four sides and angles unequal. G, If a diagonal be drawn in a parallelogram, as also two lines parallel to the sides, through the same point of the diagonal, the parallelogram will be divided into four parallelograms ; and three of them, viz. one of those described upon the diameter and the two supplements (i. e. the two parallelograms, which are not described about the diameter), form a figure called a gnomon 3 tlius the three parallelograms H I L make a gnomon, as do also the three parallelograms I K L. All figures having more than four sides, are called polygonals or multi laterals. OF CURVES OR CURVILINEAL FIGURES. A, a circle, is a surface or figure perfectly round, described upon a centre, fronx which the circumference in all its parts is equally distant. abed, a circumference. As the extremity of a circle, or it is tlie circular line that bounds it. B, an oval, is a curvilineal figure described upon several centres, and divided into two equal parts by all its diameters. C, an ellipse, is also a curvilineal figure described upon several centres in the shape of an egg, and has but one diameter that divides it into two equal parts. D, a volute or scroll, is a figure or surface bounded by a spiral line. E is a cylindric surface. F is an irregular curvilineal figure, composed of several dissimilar curve lines. f3 n L -y? ^ I Pie 15 OF COMPOUND FIGURES. At A SEMICIRCLE, is 3. figiiTe Contained between half tlie circumference and the diameter. B, a portion of a circle, is a figure comprehended witliin any part of a circle, and a right line. f, a large portion of a circle, is greater than half tlie circle. g, a small portion of a circle, is that which is less than half the circle. C, a sector, is a figure contained between two semidiameters, and an arc, greater or less than a semicircle. There is also a large or small sector. D, concentric fignires, are such as have the same centre. 3 J eccentric figures, are such as are described upon ditTerent centres. OF REGULAR AND IRREGULAR FIGURES. A, a regular figure, is that which has its opposite parts similar and equal. B, an irregular figure, is such a one as is composed of angles and sides that are dissimilar. E E, similar figures, are such as have all their sides proportional, though one may be greater, equal, or less, than another. F F, equal figures, are such whose contents are equal, though tliey may be similar or dissimilar. C, an equiangular figure, has all its angles equal. E E, o?ieJigure is said to le similar or equiangular to another, when all the re^ spective angles of the one are equal to all the respective angles cf the other. C D, an equilateral figure, is one that has all its sides equal. G G, similar curvilineal figures, are such as will admit similar polygons to be inscribed in them, or circumscribed about them. THE AXIOMS, •• i c 2 18 AXIOMS. I. Things equal to the same Third, are equal to one another. The lines A C, A C, which are equal to A B^ are also equal to one another. II. If to equal Things, equal Things le added, the whole will be equal. The lines A C, A C, are equal. The lines CD, CD, added are equal. The whole AD, AD, are also equal, III. If from equal Things, equal Things be taken away, the Remainders will be equal. If from the equal lines . . AD, AD you take away the equal lines . A C, A C the remaining parts . . C D, C D will be also equal. IV. ff to unequal Things, you add equal Things, the whole will be unequal. If to the unequal lines , . D E, D E you add the equal lines . . A D, A D the whole . . . . AE, AE will be unequal. V. If from unequal Things, equal Things be taken, tJie Remainder tvill be unequal. If from the unequal lines . . A E, A E you take away the equals . . A D, A D the remainders . . , D E^ D B will be unequal. VI. Things double the same Third, are also equal to one another. The right lines . . . D D, D D that are double the line , , J\,T> are equal among themselves. VII. Things, that are Halves of the same, or equal Things, are also equal. I'he lines . . . . AD, AD which are halves of the lines . D D, D D are equal to one another. ^Vhat has been said of lines^ may also be said of ftumbers^ surfaces^ and Lodies. 2 '9 Fl7 THE PETITIONS 21 THE PETITIONS OR DEMANDS. PETITION I. Draw a right line from the pQint ,,,,,, A to the point ......•..•£ OPERATION. Apply a ruler to the points . . • A & B Draw the line demanded ... A B by carrying the pencil along the ruler, and close to it from the point . , A to the point .... B PETITION II. Produce infinitely the line . .. • . , . CD on the side of the extremity ...... /> OPERATION. Join the ruler to the line . , . .CD Continue infinitely that line . , .CD on the side of the extremity ^ . D by carrying the pen along close to the ruler towards ..... E PETITION III. Descrile a circle upon the point ....... A and at the distance . . . . . • • . AB OPERATIOlSr. Set one of the points of the compass upon the given point .... A Open the other to the given point . , B Turn the compasses about upon the point . A and trailing the point . , , . B draw the circle demanded . . B^C D PETITION IV. On the points , . , E, ^F mtLlie an intersection or section. OPERATION. Open the compasses at discretion, but so that the distance of the two points of the compasses may be greater than half the distance of the points proposed E &F Wilh this distance of the compasses Upon the point E describe the arc * , . , • . LM upon the point F draw the arc . . . . . . HI and the intersection required will be . , • . • G Z2 P18 B 7^^^-- ''''TJf yERSIT Y THE FIRST BOOK OF THE DESCRIPTION OF LINES, 24 BOOK THE FIRST. PROPOSITION I. To erect a Perpendicular upon the Middle of a right Line. POSITION. Let C be the point proposed in the middle of the Hne A B^ upon which the perpendicular is to be erected. OPERATION. Upon the given point describe at pleasure the semicircte Upon the points , make the section From the point draw the line demanded through the section C DE D&E I C CO I This line C O will be perpendicular to the line given A B^ and erected upon the point proposed C. PROPOSITION II. To erect a Perpendicular upon the Extremity of a right Line proposed. Let A be the extremity proposed of the line A B, upon which the perpondi- cular is to be erected. OPERATION Take at pleasure the point above the line from that point with the distance Describe the portion of the circ' Draw the right line through the points Draw the line demanded it will be perpendicular to and at tlie extremity proposed ANOTHER WAY. Upon the point A describe the arc Upon the point g describe the aj"c Upon the point h describe the "arc Upon the point m describe the arc Draw the line required C AB C CA EAD DCE D8fC AE AB A ghm Ah Amn hn A n Zf A- /\ ^^s , I n L.^ K^ / \ / "•■.. \c \ \ i . "^ J, M zi PI- 10 JV v^ -'■<:. ^^>^fefe#J^'^"^^ 27 PROPOSITION III. Upon an Angle given to erect a right Line that inclines neither to the right Hand nor to the left. Let B A C be the angle upon which tlie right line is to be raised, that inclines neither to the right hand nor to tlie left. OPERATION. Upon the angle given A describe at pleasure the arc BC upon the extremities . B&C make the section D from the point of the angle given A draw the line required . • AD through the section . . • D This riglit line AD shall be erected upon the angle BAG without inclining either to the right or left. PROPOSITION IV. To let fall a Perpendicular upon a given Line, from a Point without the Ling. Let C be the point from which a line is to be let fall perpendicular to A B. OPERATION. Upon the given point . , C describe at pleasure the arc DE cutting the line AB in the points . D&E upon those points . D&E As centres make the section • F draw the line CF and the line CO will be the line required. 28 PROPOSITION V. Through a given Point to draw a Line parallel to a given right Line. Let A be the given point through which a line is to be drawn parallel to the line B C. OPERATION. Draw at pleasure the oblique line « AD upon the point • A Describe the arc . D£ upon the point • D Describe the arc AF make the arc , DG equal to the arc , AF Draw tlie line required . MN through tlie points . • A&G OTHERWISE. Upon the centre A describe the arc , EFG touching the line , BC TFithout altering the Legs of the Compasses. Upon the point H describe the arc . L R I The point H is taken at pleasure in the line B C Draw the line demanded . . OP through the point ... A and touching the arc . . L R I PROPOSITION VI. To bisect a given Jinite right Line. POSITION. Let A B be the right line proposed to be divided into two equal parts. OPERATION. Upon the extremity ... A as a centre^ describe tlie arc . . CD Without altering the Distance of the Legs of the Compasses. Upon the other extremity . , B as a centre describe the arc . . E F These Arcs are to he made so as to intersect each other. Draw the right line . , , G H through the intersections . . G & H A:B tlien will be bisected at the point . O V Flu M N B X>,X F ■ C o A R P B '-'G / ■■■■1 B -.F .--"' li H c t VI CTx^ I ..:-E «-fN oi \ / •■■' » I'^l^mrRSlTV .U-'^'^ Or ' VR 30 Tl-iz ^^'^ 31 PROPOSITION VII. To bisect a given rectilineal Angle, Let B A C be the angle proposed to be bisected. OPERATION. Upon the angular point A describe at pleasure the arc . DE upon the points D&E As centres make the section O draw the line AO This line AO will divide the given angle . BAG into two equal parts. PROPOSITION VIII. At the End of a given right Line to make a rectilineal Angle equal to a given rectilineal Angle. Let A be the end of the line A B, at which an angle is to be made equal to a given rectilineal angle , . , . . C D G OPERATION. Upon the angular point . , D describe at pleasure the arc . . " C G Without altering the Opening of the Compasses. Upon the extremity describe the arc Make the arc equal to the arc draw the line The angle will be equal to the angle which was the tiling proposed d2 A HO HE CG AE BAE CDG 3Z PROPOSITION IX. To divide a given right Line into any Numher of equal Parts required. Let A B be the line proposed to be divided into six equal parts. OPERATION. From the point A draw at pleasure the line AC through the extremity B Draw the line BD parallel to the line AC from the points . A &B and along the lines AC, BD Cany any six equal parts. viz. efghIL along the line AC R q p 0 n m along the line BD draw the lines, en, f o. gP^ hq. IR then the line , AB will be divided into six equal parts at the Sections . S,T,V,X,Y PROPOSITION X. To draw a Tangent to a Circle proposed through a given Point, Let A be the point through which the tangent to the circle D O P is to be di'awn. OPERATION. From the. centre of the circle . . B draw the secant . . . B A divide the line . • . B A into two equal parts in . , C upon the point . . . C with the radius . • . C A Describe the semicircle . . A D B cutting the circle in . . . D from the given point . • A Draw the right line . . . AB through the point . , . D This right line . . , AB will be the tangent required. DC 33 Pi (?> G. H -\ \ G C F .-"-\ \ \ \ Ji ..- '^ .,..--\S \T W \X •••7 ^^] • \ \ '> '• \ \ j-'' R \ \ \ \ V^-"'-Q \ \ \ V-"P \ \ \..--'o \.-k-'^^ (T W;.V..|{03 A'rv^^^'O;. XL 34 Ph4 :kr ■■•••■--- 1 35 PROPOSITION XI. To draw a right Line that shall be a Tangent to a Circle at a given Point, Let A B C be the given circle, and the point of contact in its circumferenee A. OPERATION. From the point or centre D draw the hne DF through the point proposed . A Through the point proposed A and to the line DF draw the perpendicular AH continued towards I This tangent . HI will touch the circle at the point A which was tlie thing required. PROPOSITION XII. j4 Circle and a right Line that touches it, being given, tojind the Point of Contact Let A B C be the circle to which the line G H is a tangent. OPERATION. From the centre of the circle . , F let fall the perpendicular . . F C upon the tangent . . . D E The section . . . . C will be the point of contact sought. 36 PROPOSITION XIII. To draw a spiral Line about a ginen right lAnc. Let I L be the line about which the spiral line is to be described. OPERATION. Divide half the right line . . I L into as many equal parts as there are to be revo- lutions. EXAMPLE. To make one of four Revolutions, Divide the half Bi into four equal parts BCEGI Divide also BC into two equal parts in A upon the point A Describe the semicircles BC, DE, FG, HI upon the point B Describe the semicircles CD, EF, GH, IL and you will have the spiral line sought. PROPOSITION XIV. Between two given Points to find two others directly interposed. Let A & B be the points given, between which two others are to be found directly interposed, by the help of which a right line may be drawn from the point A to the point B, with a short mler. OPERATION. Upon the points . . . A & B as centres, make the intersections . C & D upon the points . . . C & D As centres make the intersections . G & H These points . . . G & H are the points required, by the assistance of which a right line may be drawn from the point A to the point B, which could not be done at once with a rule less than the length between . . , . , A B 5^01 ^7 Fl. i5 LlH Cx ll SV A A y ■lA-.. UNIVERSITV THE SECOND BOOK. OF THE CONSTRUCTION OF PLANE FIGURES, 8tji BOOK THE SECOND. PROPOSITION I. To make an equilateral Triangle upon a given Line. Let A B be the given line upon which the equilateral triangle is to be con- stmcted. OPERATION Upon the extreme point A with the radius • AB Describe the arc BD upon the extremity B with the radius BA Describe the arc AE from the intersection C Draw the lines CA, CB ABC will be the equilateral triangle required. ^PROPOSITION II. To make a Triangle whose three Sides are equal to three given right Lines. Let A B C be the three given lines j a triangle is to be made whose three sides are eqaal to them. OPERATION . Draw the right line DE equal to the line AA upon the point D with the radius • BB Describe the arc GF upon the point E with the radius CC Describe the arc HI from the intersection O Draw the lines . , OE, OD The trianale . DEO will be composed of three sides equal to the three lines given . A A, B B^ C C ^0 ri-id CBA JL J OR'>i\i m 42 PROPOSITION III. To make a Square upon a given right Line. Let A B be the given right line^ upon which the square is to be made. OPERATION. Erect the perpendicular AC upon the point A As a centre, describe the arc BC upon the points . B&C with the radius AB Make the section D from the point D Draw the lines DC, DB A BCD will be the square required to be constructed upon the given right iine . . . , , • , . AB PROPOSITION IV. To make a regular Pentagon upon a given right Line. Let AB be tlie given line, upon which the pentagon is to be constructed. OPERATION. Upon the extremity %. A and Vv'ith the radius . • • AB Describe the arc ' BDF Erect the perpendicular AC Divide the arc BC into five equal parts ID , LM Draw the right line AD Divide the base A ii into two equal parts in O Erect the perpendicular OE upon the intersection E with the radius EA Describe the circle ABFGH Carry round five times, the line . AB in the circumference of the circle, aqd a regular equiangular equilateral pentagon will be com- pleted. £ 2 43 PROPOSITION V. To make a regular Hexagon upon a given right Line. Let A B be a right line, upon which a regular hexagon is to be made. OPERATION. Upon the extremities and with the radius Describe the arcs upon the section Describe the circle carry six times the line given in the circumference, and you will have a regular hexagon . . ABEFGD upon the given line • . . A B . A&B AB AC, BC C ABEFG AB PROPOSITION VI. Upon a given right Line to describe any Polygon from an Hexagon to a Dodecagon. Let AB be a line upon which an hexagon, heptagon, or octagon, &c. is to be made. OPERATION. Bisect the line A B in the point . O Erect the perpendicular . . O I upon the point B describe the arc . AC Divide A C in six equal parts M, N, P, Q, R This is to he done, if an Heptagon is to be made. Upon the point C with the interval of one part , . .CM describe the arc . • . M D D will be the centre for describing a circle capable of containing seven timeff the line given. For an Octagon. Upon the centre C, with the interval of two parts . • . C N Describe the arc . . . N E E will be the centre of a circle capable of con- taining eight times the given line • AB For an Heptagon. Take three parts . , • C P And so for the rest, adding one part. u fiw YI / / / / _., ■''''-' -:rS A I B vnr f/^-N-..., ,; ,f> vvy-::^^ 46 PROPOSITION VII. To make a Polygon of any Number of Sides from twelve to twenty-four j upon a given right Line. Let A B be the line, upon which the polygon is to be made. OPERATION. Divide the arc . •• .AC into twelve equal parts from the point C Take as many of the parts of . . C A as the number of the sides of tlie polygon is above twelve. EXAMPLE. If you would describe a Polygon of ff teen Sides. Upon the point . . . C with the radius of three of these parts C E describe the arc • . . E O AC of twelve, C 0 of three together, make fifteen. Upon the point O with the radius . O B describe the arc . • , B F Upon the point F with the radius , F A describe a circumference, and it will contain the line given . . . . A B fifteen times. And so also for any other Polygon. PROPOSITION VIII. To describe a Portion of a Circle capable of containing an Angle equal to an Angle given upon a given right Line. Let A B be the right line, upon which a portion of a circle capable of con- taining an angle equal to the given angle, is to be described, C. OPERATION. Make the angle . • .BAD equal to the angle . • . C Erect upon . , , , A D the perpendicular . . . A E Bisect the line , , . A B in the point . . . H Erect the perpendicular • . H F upon the section • • . F with the radius . , .FA Describe the portion of the circle . A E B All the angles you make in this segment of the circle, and upon the given line . A B will be equal to the angle • . C 47 PROPOSITION IX. Tojind the Centre of a given Circle. Let A B C be the circle proposed, whose centre is to be found. OPERATION. Draw at pleasure the right line . . A B terminating in the circumference . ABC Bisect the right line , . . A B by the line . . . , D C Bisect also the right line . , C D* in the point . . . F The point F will be the centre of the circle required . , , .ABC . PROPOSITION X. To complete the Circumference of a Circle whose Centre is lost. Let A B C be the part of the circumference given, whose centre is to be found, in order to the finishing the circle. OPERATION. Take at pleasure the three points . ABC in the circumference begun upon the points A & B Make the sections . E&F Draw the right line EF upon the points . m. . B&C Make the sections . G&H draw the right line GH upon the intersection and centre I and with the interval lA complete tlie circumference begun. tx 4^ n fo w«v«*^ *■• "» -^^'^'^'Rs/ryJ XT yy-^f H m .50 PROPOSITION XL To describe a Circle that shall pass through three given Points. Let A, B, C, be the three points through which the circle is to pass. OPERATION. Upon the points given . . A, B, C describe three circles DE H, D E F, F G L with the same radius, and intersecting a'^ the points . , D&E, F&G iDraw the right lines . , D E, F G till they meet in . , .1 upon the point . . .1 with the radius . . . I A Describe tlie circle requixd. This operation is similar to the preceding. 11/ PROPOSITION XII. To describe an Oval upon a given Length. Let A B be the length upon which the oval is to be made OPERATION. Divide the length given AB into three equal parts ACDB upon the points . CSrD with the radius CA Describe the circles . AEF, BEF upon the intersections . E&F . and with the diameter EH As a radius describe the arcs IH, OP A I H B P O will be the oval required. 51 PROPOSITION XIIL To describe an Oval upon two given Diameters. Let A B;, CD, be the diameters upon which the oval is to be constructed. OPERATION* Make the ruler . ♦ .MO equal to the great semidiameter . A E upon which mark the length . M N equal to the lesser semidiameter . C E This Ruler being thus disposed^ Place it after such a manner upon the diameters AB, CD that the point ♦ . . N sliding along the line the extremity- may always be in the line carrying along thus the rule Describe the oval with the extremity AB O CD MO M PROPOSITION XIV. To find the Centre and the two Diameters of an Oval. Let A B C D be the oval proposed, whose centre and diameters are to be found. OPERATION. In the oval proposed ABCD draw at pleasure , the two parallel hnes ANHI Bisect the lines ANHI in the points . L&M Draw the line , PLMO Bisect it in . E and the point E will be the centre upon the point E Describe at pleasure the circle . Foa cutting the oval in . , . F&G through the intersections F&G Draw the right line FG Bisect it in . R Draw the greatest diameter BD through the points ER through the centre E Draw the least diameter . AEC parallel to the line FG and what was proposed will be effected xnr rr^t xn^ Yr>J-^nr, p^;. n.23 D E r^^SC^-tKC^-^ ^^t^^^^^^ ^riPy„ tu^^vv; XT ^7 ^/ ^ I t BL A PROPOSITION XI. To inscrile a Square in a given Triangle, Let A B C be the triangle in which the square required is to be inscribed. OPERATION. Erect the perpendicular AD upon the extremity of the base AB Make this perpendicular AD equal to the base AB from the angle C Draw the line CE parallel to the line AD Draw the oblique line , DE through the section F Draw the line FG parallel to the base AB Draw the lines FH, GI parallel to tlie hue CE And the square required will be FGHI PROPOSITION XII. To inscribe a regular Pentagon in an equilateral Triangle. Let A B C be the triangle in which tlie pentagon is to be inscribed. OPERATION. Let fall the perpendicular upon the centre Describe the arc Divide into five equal parts the arc Carry on the sixth Draw the line Divide into two equal parts Upon the point describe the arc Draw the right line Make the part equal to the part Draw the right lines upon the centre with the distance of the section Describe the arc upon the points Describe the arcs Draw the lines And then tlie pentagon demanded w Al A BIM BI IM AM AM L A LD LDtoH AG BH G, MC D N NO NO DQ, DP OP,PQ,Na illbeDOP.QN 69 PROPOSITION XIII. Tb inscribe an equilateral Triangle in a Square. Let ABC D be the square in which the equilateral triangle is to be inscribed. OPERATION. Draw the diagonals • upon the centre and with the distance Describe the circle upon the point with the distance Describe the arc Draw the right lines Draw the right line The equilateral triangle required is AC, BD E EA ABCD C . CE GEF AF, AG . HI A HI PROPOSITION XIV. Toinscrile an equilateral Triangle in a Pentagon, Let A B C D E be the pentagon in which an equilateral triangle is to be inscribed. OPERATION. Circumscribe the circle ABCDE upon the point A and with the distance of the radius AF Describe the arc FL Cut that arc FL into two equal parts in N Draw the line FNI upon the point A with the distance . AI Describe the arc . , HOI draw the lines AH, HI The triangle demanded will be * AHI xm p XIV Plzg XV PI -50 iC-\. .,«•-■. n PROPOSITION XV. To inscribe a Square in a Pentagon, Let A B C D E be the pentagon in which a square is to be inscribed. ' OPERATION. Draw the line BE let fall the perpendicular ET from the extremity of BE Make this perpendicular ET equal to the line BE Draw the line AT through the section O Draw the hne . . . OP parallel to the side CD On the extremities O&P erect the perpendiculars OM, PN Draw the line . . NM The square required will be . NMOP BOOK THE FOURTH. OF THE CONSCRIPTION OF FIGURES. PROPOSITION I. To circumscribe a Circle about a given Triangle, Let A B C be the triangle about which the circle is to be circumsc];ibed. OPERATION. Describe the circumference . , ABC through the three points . . A^ B, C and the thing required will be done. 62 73 PROPOSITION II. To drcumscrile a Circle about a Square. Let A B C D be the square about which the circle is to be circumscribed. OPERATION. Draw the two diagonals . A B, C D upon the intersection or centre • G with the distance Describe the circle demanded GA ABCD PROPOSITION III. To drcumscrile a Triangle similar to a given Triangle, about a given Circle, Let D E V be the circle, about which a triangle, similar to the triangle F G H, is to be described. OPERATION. Draw the diameter . . . A B through the centre c Make the angle . ACE equal to the angle H Make the angle . BCD equal to the angle G Produce the lines E C, D C towards . R&S Draw the tangent NO parallel to the line DR Draw the tangent OI parallel to the line ES Draw also the tangent . NI parallel to the diameter AB I N 0 will be the triangle required similar to the triangle F G H, and cir- cumscribed about the circle D E V. ir 1 B Tl^3i in: 0oTe l)b«^ '^ Of tH6 , '?> UNIVERSITY ^ 7^ Pi- St 76 PROPOSITION IV. To circumscribe a Square about a Circle. Let A B C D be the circle about which a square is to be circumscribed. OPERATION. - Draw the diameters . . A B, C D intersecting each other at right angles in O upon the points with the distance Describe tlie semicircles . A, C, B, D AO HOG, HOE EOF, FOG Draw the right lines E F, F G, G H, H E through the intersections E, F, G, H The square demanded will be E F G H PROPOSITION V. To circumscribe a Pentagon about a given Circle. Let A B CD E be the given circle about which a pentagon is to be circumscribed. OPERATION. Inscribe the pentagon . . A B C D E upon the centre . . . F and through the middle of each side Draw the lines . F O, F P, F Q, F R, F S uidw Liie iiiic • • Draw the tangent . pa through tlie point A Upon the centre F with the radius . FP Describe the circle OPQRS Draw the sides of the pentagon demanded through the sections . O, P, Q^ R, S n PROPOSITION VI. To circumscribe a regular Polygon about another of the same Sort. Let B C D E F G be the polygon given, about which another similar polygon is to be circumscribed. OPERATION. Produce two sides as • • BG, EF until they meet in , . H Draw the line * AH Draw the line , EI bisecting the angle , GFH upon the centre . A with the distance . AI Describe the arc . , IMO Draw the radius's ALAM, AN, AG through the middle of each side. Draw the sides of the exterior polygon demanded, through the sections I, L, M, N, O, P PROPOSITION VII. To circumscribe a Square about a given equilateral Triangle, Let A B C be the equilateral triangle, about which a square is to be circum- scribed. OPERATION. Bisect the base BC in the point E Produce the base BC both ways towards D&D Make the lines ED&ED equal to the line EA Upon the point E with the distance EC Describe the semicircle BFC draw the line AEF From the point F draw the lines FCG&FBG and the squai'e required will be AGFG 7S vr ;*H PI 3,1 <>^ v^ of "^ VBT- /J '9 n.m xs. *"Tr 80 PROPOSITION VIII. To circumscrile a Pentagon about an equilateral Triangle, Let A B C be the triangle given, about which a pentagon is to be circum- scribed. OPERATION. Upon the points or angles , A/B,C and with the same opening of the compasses, de- scribe at pleasure the arcs DE, LP Divide the arc DO into five equal parts 1,2,3,4,5 upon the centre or section o And with the distance of four parts . O N describe the arc NME Draw the right line . AE F Cut off the arc , MP equal to the arc . EN Draw the right line fPcg equal to the line fA Make the arc DH equal to the arc DE Draw the sides AI,IR equal to the sides . . Af, fG The side IR will complete the pentagon demanded. PROPOSITION IX. To circumscrile a Triangle similar to a given Triangle, about a Square. Let D E F G be the square about which a triangle is to be circumscribed simi- lar to the triangle ABC. OPERATION. Make the angle EFM equal to the angle . A Make the angle MEF equal to the angle . B Produce the lines . ME, MF, DG towards I&H HIM will be the triangle required, similar to the triangle . . . . ABC and circumscribed about the square D E F G 81 PROPOSITION X. To circumscribe a Pentagon about a Square. Let A B C D be the square about which a pentagon is to be circumscribed. OPERATION. Produce the side CB towards N Bisect tlie side AB in the point Erect the perpendicular upon the points with the distance • * R RV . B, D, C BR Describe the arcs RN, ST, ST Divide the arc RN into five equal parts Make the angle RH, GF, EN RBV with the distance of two parts RG Make the angles SCT^SDT with the distance of one part RH Produce the lines VB, CT, toO Make the line oa equal to the line . , . O V Draw the other sides after the same manner^ and you will have the thing required BOOK THE FIFTH. OF PROPORTIONAL LINES. PROPOSITION I. Tojind a Mean proportional between two given Lines. Let A & B be tlie two lines between which a mean proportional is to be found. OPERA-TIOIf. Draw an indetermined line GH Make CE equal to the line A Make ED equal to the line . • B Bisect . . CD in the point . . , I upon the point . . , I and with the distance IC Describe the semicircle CFD Erect the perpendicular EF This line . . . . EF will be a mean proportional between A&B X e^ Fl 36 T^ - \ ic« t^^ %^ # n Fise P---I- 1 \^ 1 / \ 1 / \ f \ 1 B G nr D/ B Nf.«.^^ie«i 84 PROPOSITION II. Given the Sum of the Extremes and the mean Proportional, to distinguish the Means. Let AB be the sum of the extremes (i. e. the two magnitudes connected without any distinction), and C the mean proportional, by whose assistance the pointj where the extremes join, is to be distinguished. > OPERATION. Bisect the hne AB in the point G upon the point G with the interval GA Describe the semicircle AEB Erect the perpendicular BD equal to the mean proportional C Draw the line DE parallel to the line AB from the section E Draw the line EF parallel to the line BD Then will the point where the extremes ; join be F so that C or its equal EF shall be a mean proportional between A F & B F PROPOSITION III. Given the Mean of three Proportionals and the Difference of the Extremes, to find the Extremes, Let G H be the mean proportional, and A B the difference of the extremes, required the length of the extremes. OPERATION. Erect the perpendicular BC at the extremity of the difference AB and equal to the mean GH Bisect the difference AB in the point D Produce both ways towards . E&F upon the point D with the distance DC Describe the semicircle . ECF The extremes required will be BE, BF 85 PROPOSITION IV. Two right Lines being given, to find a third Proportional. A B, AC, are the two given right lines, to which a third proportional is to be found. OPERATION. Make at pleasure the angle DNE cut off the part NH equal to the line AB Cut off the part NO equal to the line . AC Draw the line HO Draw the line DE parallel to Uie line HO E O will be the third proportional required. PROPOSITION V. To cut off' from a given Line, a Part that shall be a mean Proportional between what remains and another given right Line. Let A A be the line, of which a part is to be cut off, that shall be a mean proportional between what remains and the line proposed BB. OPERATION. Draw the indefinite line . . CD cut off the lines . . D E, E C equal to the lines . , A A & B B Describe the semicircle . , . C F D Erect the perpendicular , . E F Bisect the lines , , . C E in the point . . , B upon the point , , , B with the distance , . , B F Describe the arc . , . F G Cut off the part demanded . . . AH equal to the part . . . EG A H will be the mean proportional between the remainder . . , , HI and the other line proposed , , BE I\ ^/ PI^3 ^i: V GA I \ j \ i \ ! \ E B DGE ^ UNIVERSITY VI rl s 4 . H.< D E 3 -^(T .^ zn CJftjL I 2 VJT a •> /■ / 4 ' yd y A I F ,.. - ' E { H- 88 PROPOSITION VI. Tojind a fourth Proportional. A, B, C, are three equal lines proposed; a fourth is to be found, which will be to the third just as the second is to the first. OPERATION. Make at pleasure the angle . GDH cut oif the part DE equal to the line A Cut otf the part DF equal to the line B Cut ofr the part EG equ:d to the line C Draw the line EF Draw the line . . . GH parallel to the line . EF F H will be the fourth proportional demanded. PPxOPOSITION VII. Tojind two mean Proportionals between two given Li?ies. Let I & H be the lines proposed, between which two mean proportionals are to be found. OPERATION. Draw the line A B equal to the 'line H Let fall the perpendicular BC equal to the line I Draw the line AC Bisect the line AC in the point F Erect the perpendiculars AG, CR upon the point or centre F Describe the arc DE so that the cord DE may touch the angle B AD, C Ej will be the mean pre )portionals between the given lines - . l&H 11 2 S9 PROPOSITION VIII. To cut two given Lines j each into two Parts, so as that the four Segments may le proportional. A B;, AC, are the lines proposed to be cut according to the proposition. OPERATION. Make the right angle Cut off the hne equal to the line Cut off the line equal to the line Draw the hypothenuse Describe the semicircle from the section Draw the line parallel to the line and the line parallel to the line A B will be cut in O C also in so that B E will be to as E D to to D F, as D F is to BOC BO AB OC AC BC BDO D DE CO DF EO E F D DF,&ED FC PROPOSITION IX. The Exx-e.'-s of the Diagonal of a Square above the Side, being given, to find. its Side. Let A B be the excess of the diagonal of a square above its side, to find its macfoitude. OPERATION. Erect the perpendicular BC equal to the excess AB Draw the line . AC produced towards D upon the point . , C and with the distance CB Describe the arc BD A D will be the side of tlie square ■ A the excess . . . . AB of whose diagonal AE above tlie said side . • AD VUL <:io • E- B E D C F nj^o D lO B B -A I^ ^^?Y/lV.^1ib ^^Ufi^^ OF THE UNIV£«S!TV h X Q1 n 4-0 H B^ •pv- 1 "^\.^ \ >''' .--^J,— ,-" r^'"' 1 , *', 1 ~^^ 1 1 ,' ^^^v ^-^^ \ i ""\j \^ ± ^-^ ^^ ""-. -.^ ""^i( D JI d- A$ %i B- r^-^? I eJc -.^ F .H ^f-./ :n^ o fV,i/,^/»t, ^^03 92 PROPOSITION X. To cut a given finite Line in extreme and mean Proportion. Let A B be the line that is to be cut, so that the rectangle of the whole line, and one of the parts, may be equal to the square of the other. OPERATION. Erect the perpendicular produce it towards Make equal to half Upon the point and with the distance Describe the arc upon the point with the distance Describe the arc The line will be cut in the point in the proportion required : for if you make the rectangle A h of the whole A B and part B E, it will be equal to the square A f made upon the other part . . . . A E AD C AC AB C CB BD A AD DE AB E PROPOSITION XI. To divide a given right Line in any Ratio proposed. Let A B be the line proposed to be divided according to the ratios of C^ D, E, F. OPERATION. Upon the point or extremity Draw at pleasure the line Make equal to tlie line or ratio Make equal to the line Make equal to the line Make equal to the line Draw the line Draw the lines parallel to the line A AG AH C HI D IL E IM F BM LN, lO, HP BM The line A B will be divided in the points P, O, N according to the ratio demanded. 93 PROPOSITION XII. To make upon a g'wen right Line two Rectangles, that shall be in any given Ratio to one another , Let A B be the line upon which two rectangles are to be made, which shall be to one another as C to D. OPERATION. Divide the right line AB at the point . E in the ratio of C toD Make the square ABHF Draw the line EI parallel to the line AF B E I H, A E I F, will be the rectangle required. The rectangle AI is to the rectangle EH as the line D is to the line C -■^4 xn ;• 1 H ^ \ \ \ /^\^ / /^ . / / c '. _, y' ~-v ', I. / ^-. fl B ^-^ ^'J" XIII Pi 4Z mz m/ c c *». i / 1 it , ,. C ^ _ V J / c h d 1 ^ j 1 I I t i t * 1 3 Scale, of IncAjbs m3^ 95 PROPOSITION XIII. To make a geometrical Plan and Elevation. The subject fixed on in this plate is a tea-caddy. No. 1 Shews the geometrical plan. 2 The geometrical elevation. 3 The tea-caddy open in perspective. OPERATION. Make a scale of inches the size you wish to introduce on paper, measure the length of the tea-caddy with a common ruler, then take as many parts of your scale as it measures. Set it on the line ..... Measure the side of the tea-caddy, and fix it upon the line With the number of parts answering to the measurement. Measure the canister .... Measure the sugar-glass , ... Make the line f f parallel to ... Take the height of the tea-caddy open, and make the line Raise the perpendicular on each side Take the width e d, and mark it upon the line You will then have your geometrical plan. No. 1, your geometrical eleva- tion, No. 2, with the lid open, as seen in No. 3 , designed purposely to shevr the difference of a geometrical plan and the same subject in perspective. dd de, de be gg^ gg abc hh go PROPOSITION XIV. Containing a double Cross. No. 1 is seen in perspective. 2^ The geometrical plan. 3, The geometrical elevation. No. 3, aaaa represents the centre of the elevation. No. 2, aaaa the geometrical plan of a a a a. c c c c the geometrical plan of the elevation c c c c. b b b b the geometrical plan of the cross beam b b b b. XIV CUi ^ ~^ 7 7 e 6 • ' « ^TetiBfir:^ .V- OF THE , '-V OF *^**«ii£dtJ-f££^ #._ - 4: :♦♦ X\A (,iii PI 44 '^ ^{))^^nt, IS 03 PROPOSITIOTSI XV. To make an Oval of any given Dimension, Let AB CD be the given size. Draw the two centres . . E E F 1^ CC AD c c Draw the semicircle . • • ^ Upon the line Divide the remainder of the line A D in three equal parts, . . • .1,2,3 Take two of those parts, and mark it on each side of the centre . • • "- to . . . . GG Open the compass to G G, and make the mtersec- tion F F, draw the four radii F G q, F G q. With the compass open G E, G E, draw part of the circle . . ' qEq, qEq With the compass open to F q, join the circle q q q q THE END. ^^ OF THE ^y\ S. GoswELL, Printer, Little Queen Street, Holborn. ► THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW °^^^ AN INITIAL pliFop 25 CENTS WILL BE AScsFooc-r^ r- ^■C'lNlb WILL INCREASE TO SOcL^^J^^ PENALTY DAY AND TO »I.OO ON T^."^ ™= "O""™ OVERDUE. "^ SEVENTH DAY WV-S4-^ RECEIVED jon^-^s^^ — ftPR-fr^4995— ATICN DEPT. 'MS^tt-nm-it-^ ^yN-0-e-2floa- "^^'^'^^^^^r^ Jr^AN_t2i£PT ti^Z^ZZ Jyt-4n992- 7; 1 i< ij Cipr;n ■ -:ni. lAToFigsr LD 21-100m-7. 33 ii»/»S/ ^'SftARIEs ^03'^t33744